Nuprl Lemma : nerve_box_edge1_wf

`∀[G:Groupoid]. ∀[I:Cname List]. ∀[J:nameset(I) List]. ∀[j:nameset(J)]. ∀[x:nameset(I)]. ∀[i:ℕ2].`
`∀[box:open_box(cubical-nerve(cat(G));I;J;x;i)]. ∀[y:nameset(I)]. ∀[c:{c:name-morph(I;[])| (c y) = 0 ∈ ℕ2} ].`
`  nerve_box_edge1(G;box;x;i;j;c;y) ∈ cat-arrow(cat(G)) nerve_box_label(box;c) nerve_box_label(box;flip(c;y)) `
`  supposing (∀j'∈J.j' = j ∈ Cname)`

Proof

Definitions occuring in Statement :  nerve_box_edge1: `nerve_box_edge1(G;box;x;i;j;c;y)` nerve_box_label: `nerve_box_label(box;L)` cubical-nerve: `cubical-nerve(X)` open_box: `open_box(X;I;J;x;i)` name-morph-flip: `flip(f;y)` name-morph: `name-morph(I;J)` nameset: `nameset(L)` coordinate_name: `Cname` groupoid-cat: `cat(G)` groupoid: `Groupoid` cat-arrow: `cat-arrow(C)` l_all: `(∀x∈L.P[x])` nil: `[]` list: `T List` int_seg: `{i..j-}` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` member: `t ∈ T` set: `{x:A| B[x]} ` apply: `f a` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` or: `P ∨ Q` nameset: `nameset(L)` l_member: `(x ∈ l)` exists: `∃x:A. B[x]` cand: `A c∧ B` select: `L[n]` nil: `[]` it: `⋅` so_lambda: `λ2x y.t[x; y]` top: `Top` so_apply: `x[s1;s2]` false: `False` coordinate_name: `Cname` int_upper: `{i...}` guard: `{T}` int_seg: `{i..j-}` squash: `↓T` lelt: `i ≤ j < k` and: `P ∧ Q` nat: `ℕ` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` implies: `P `` Q` not: `¬A` prop: `ℙ` cons: `[a / b]` subtype_rel: `A ⊆r B` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` bfalse: `ff` iff: `P `⇐⇒` Q` null: `null(as)` rev_implies: `P `` Q` true: `True` nerve_box_edge1: `nerve_box_edge1(G;box;x;i;j;c;y)` name-morph: `name-morph(I;J)` bool: `𝔹` unit: `Unit` uiff: `uiff(P;Q)` bor: `p ∨bq` sq_type: `SQType(T)` bnot: `¬bb` so_lambda: `λ2x.t[x]` so_apply: `x[s]` decidable: `Dec(P)` sq_stable: `SqStable(P)` open_box: `open_box(X;I;J;x;i)` eq_int: `(i =z j)` le: `A ≤ B` less_than': `less_than'(a;b)` nequal: `a ≠ b ∈ T ` less_than: `a < b` name-morph-flip: `flip(f;y)` subtract: `n - m` l_exists: `(∃x∈L. P[x])` l_all: `(∀x∈L.P[x])`
Lemmas referenced :  nameset_wf list-cases null_nil_lemma btrue_wf stuck-spread base_wf length_of_nil_lemma int_seg_properties nat_properties satisfiable-full-omega-tt intformand_wf intformless_wf itermVar_wf itermConstant_wf intformle_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_wf btrue_neq_bfalse product_subtype_list assert_elim null_wf3 cons_wf subtype_rel_list top_wf equal_wf bool_wf ppcc-problem iff_imp_equal_bool bfalse_wf true_wf false_wf iff_weakening_equal assert_wf eq_int_wf eqtt_to_assert assert_of_eq_int extd-nameset_subtype_int nil_wf coordinate_name_wf eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int l_all_wf2 l_member_wf set_wf name-morph_wf equal-wf-T-base int_seg_wf extd-nameset-nil open_box_wf cubical-nerve_wf groupoid-cat_wf list_wf groupoid_wf nerve_box_edge_wf decidable__equal_int intformnot_wf intformeq_wf int_formula_prop_not_lemma int_formula_prop_eq_lemma sq_stable__l_member decidable__equal-coordinate_name sq_stable__le decidable__le decidable__lt lelt_wf l_exists_wf not_wf eq-cname_wf assert-eq-cname set_subtype_base le_wf int_subtype_base int_seg_subtype int_seg_cases null_cons_lemma equal-wf-base subtract_wf itermSubtract_wf int_term_value_subtract_lemma name-morph-ext name-morph-flip_wf squash_wf extd-nameset_wf name-morph-flips-commute name-morph-flip-flip groupoid-square1_wf nerve_box_label_wf decidable__assert cat-arrow_wf cat-square-commutes_wf length_of_cons_lemma length_wf_nat nat_wf not-lt-2 not-equal-2 condition-implies-le minus-add minus-one-mul zero-add minus-one-mul-top add-commutes add_functionality_wrt_le add-associates add-zero le-add-cancel length_wf select_wf nerve_box_edge_wf2 subtype_rel-equal or_wf small-category_wf groupoid-square2_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis dependent_functionElimination unionElimination sqequalRule setElimination rename productElimination baseClosed independent_isectElimination lambdaFormation isect_memberEquality voidElimination voidEquality natural_numberEquality applyLambdaEquality imageMemberEquality imageElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality independent_pairFormation computeAll equalityTransitivity equalitySymmetry independent_functionElimination promote_hyp hypothesis_subsumption addLevel applyEquality because_Cache levelHypothesis inlEquality equalityElimination instantiate cumulativity axiomEquality setEquality dependent_set_memberEquality inrFormation addEquality hyp_replacement universeEquality inlFormation minusEquality productEquality

Latex:
\mforall{}[G:Groupoid].  \mforall{}[I:Cname  List].  \mforall{}[J:nameset(I)  List].  \mforall{}[j:nameset(J)].  \mforall{}[x:nameset(I)].  \mforall{}[i:\mBbbN{}2].
\mforall{}[box:open\_box(cubical-nerve(cat(G));I;J;x;i)].  \mforall{}[y:nameset(I)].
\mforall{}[c:\{c:name-morph(I;[])|  (c  y)  =  0\}  ].
nerve\_box\_edge1(G;box;x;i;j;c;y)  \mmember{}  cat-arrow(cat(G))  nerve\_box\_label(box;c)
nerve\_box\_label(box;flip(c;y))
supposing  (\mforall{}j'\mmember{}J.j'  =  j)

Date html generated: 2017_10_05-PM-03_41_31
Last ObjectModification: 2017_07_28-AM-11_26_47

Theory : cubical!sets

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